Sharp multiscale control for high order nonlinear equations

We analyze the behavior of families $(u_{\alpha }{)}_{\alpha >0}$ of solutions to the high-order critical equation $P_{\alpha }{u}_{\alpha }={\mathrm{\Delta }}_g^k{u}_{\alpha }+\text{lot}=\mathrm{\mid }{u}_{\alpha }{\mathrm{\mid }}^{2^{\star }-2}{u}_{\alpha }$ on a Riemannian manifold $M$, with a uniform bound on the Dirichlet energy. We prove a sharp pointwise control of the $u_{\alpha }$’s by a sum of bubbles uniformly with respect to $\alpha \to +\mathrm{\infty }$, that is $\mathrm{\mid }{u}_{\alpha }\mathrm{\mid }\le C\mathrm{\parallel }{u}_{\mathrm{\infty }}{\mathrm{\parallel }}_{\mathrm{\infty }}+C{\sum }_{i=1}^N{B}_{i,\alpha }$ where $u_{\mathrm{\infty }}\in C^{2k}(M)$ and the $(B_{i,\alpha }{)}_{\alpha }$, $i=1,\mathrm{.}\mathrm{.}\mathrm{.},N$ are explicit standard peaks.